Gradient discretisation method: Difference between revisions

Revision as of 07:10, 14 November 2016

Revision as of 01:29, 15 November 2016

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All the methods below satisfy the first four core properties of GDM (coercivity, GD-consistency, limit-conformity, compactness), and in some cases the fifth one (piecewise constant reconstruction).

===[[Galerkin method]]s and conforming Finitefinite Elementelement methods===

Let <math>V_h\subset H^1_0(\Omega)</math> be spanned by the finite basis <math>(\psi_i)_{i\in I}</math>. The [[Galerkin method]] in <math>V_h</math> is identical to the GDM where one defines

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The "mass-lumped" <math>P^1</math> finite element case enters the framework of the GDM, replacing <math>\Pi_D u</math> by <math>\widetilde{\Pi}_D u = \sum_{i\in I} u_i \chi_{\Omega_i}</math>, where <math>\Omega_i</math> is a dual cell centred on the vertex indexed by <math>i\in I</math>. Using mass lumping allows to get the piecewise constant reconstruction property.

=== Nonconforming ''P''<mathsup>P^1</mathsup> finite element ===

On a mesh <math>T</math> which is a conforming set of simplices of <math>\mathbb{R}^d</math>, the nonconforming <math>P^1</math> finite elements are defined by the basis <math>(\psi_i)_{i\in I}</math> of the functions which are affine in any <math>K\in T</math>, and whose value at the centre of gravity of one given face of the mesh is 1 and 0 at all the others. Then the method enters the GDM framework with the same definition as in the case of the Galerkin method, except for the fact that <math>\nabla\psi_i</math> must be understood as the "broken gradient" of <math>\psi_i</math>, in the sense that it is the piecewise constant function equal in each simplex to the gradient of the affine function in the simplex.

=== Mixed Finitefinite Elementelement ===

The [[mixed finite element method]] consists in defining two discrete spaces, one for the approximation of <math>\nabla \overline{u}</math> and another one for <math>\overline{u}</math>.